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A258458 Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.

%I A258458 #13 Oct 15 2018 10:21:56
%S A258458 1,7,33,130,463,1557,5031,15877,49240,151116,460173,1394645,4212071,
%T A258458 12693724,38195286,114817389,344911117,1035659955,3108817911,
%U A258458 9330152740,27997803871,84008165515,252053831034,756220333901,2268778132337,6806569134920,20420175154486
%N A258458 Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.
%H A258458 Alois P. Heinz, <a href="/A258458/b258458.txt">Table of n, a(n) for n = 3..1000</a>
%F A258458 a(n) ~ c * 3^n, where c = 1/(6*Product_{n>=1} (1-1/3^n)) = 1/(6*QPochhammer[1/3, 1/3]) = 1/(6*A100220) = 0.297552056999755698394581... . - _Vaclav Kotesovec_, Jun 01 2015
%p A258458 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258458       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258458     end:
%p A258458 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258458 a:= n-> T(n,3):
%p A258458 seq(a(n), n=3..35);
%t A258458 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
%t A258458 T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
%t A258458 a[n_] := T[n, 3];
%t A258458 Table[a[n], {n, 3, 35}] (* _Jean-François Alcover_, May 22 2018, translated from Maple *)
%Y A258458 Column k=3 of A256130.
%Y A258458 Cf. A320545.
%K A258458 nonn
%O A258458 3,2
%A A258458 _Alois P. Heinz_, May 30 2015